Quartile Deviation calculator for ungrouped data

Quartile Deviation for ungrouped data

Quartile deviation (QD) is an absolute measure of spread or dispersion based on the quartiles. Quartile deviation is defined as half of the distance between the third quartile $Q_3$ and the first quartile $Q_1$. It is also known as semi-interquartile range. Thus the quartile deviation is given by

$QD = \dfrac{Q_3-Q_1}{2}$

Coefficient of Quartile deviation

The relative measure of spread or dispersion based on quartile deviation is the coefficient of quartile deviation (QD). The coefficient of quartile deviation is used to study and compare the degree of variation for two or more data set having different units of measurements.

Coefficient of Quartile Deviation is given by

Coefficient of $QD = \dfrac{Q_3-Q_1}{Q_3+Q_1}$

where

• $Q_1$ is the first quartile
• $Q_3$ is the third quartile

The formula for $i^{th}$ quartile is

$Q_i =$ Value of $\bigg(\dfrac{i(n+1)}{4}\bigg)^{th}$ observation, $i=1,2,3$

where $n$ is the total number of observations.

Quartile Deviation Calculator for ungrouped data

Use this calculator to find the Quartile Deviation (QD) for ungrouped (raw) data.

How to calculate quartile deviation for ungrouped data?

Step 1 – Enter the $x$ values separated by commas

Step 2 – Click on "Calculate" button to get quartile deviation for ungrouped data

Step 3 – Gives the output as number of observations $n$

Step 4 – Gives the output as ascending order data

Step 5 – Gives all the quartiles $Q_1$, $Q_2$ and $Q_3$

Step 6 – Gives the output of quartile deviation and coeffficient of quartile deviation

Quartile deviation for ungrouped data Example 1

A random sample of 15 patients yielded the following data on the length of stay (in days) in the hospital.

5, 6, 9, 10, 15, 10, 14, 12, 10, 13, 13, 9, 8, 10, 12.

Find quartile deviation and coefficient of QD.

Solution

The formula for $i^{th}$ quartile is

$Q_i =$ Value of $\bigg(\dfrac{i(n+1)}{4}\bigg)^{th}$ observation, $i=1,2,3$

where $n$ is the total number of observations.

Arrange the data in ascending order

5, 6, 8, 9, 9, 10, 10, 10, 10, 12, 12, 13, 13, 14, 15

First Quartile $Q_1$

The first quartle $Q_1$ can be computed as follows:

$$ \begin{aligned} Q_{1} &=\text{Value of }\bigg(\dfrac{1(n+1)}{4}\bigg)^{th} \text{ observation}\\ &=\text{Value of }\bigg(\dfrac{1(15+1)}{4}\bigg)^{th} \text{ observation}\\ &= \text{Value of }\big(4\big)^{th} \text{ observation}\\ &=9 \text{ days}. \end{aligned} $$

Thus, lower $25$ % of the patients had length of stay in the hospital less than or equal to $9$ days.

Third Quartile $Q_3$

The third quartile $Q_3$ can be computed as follows:

$$ \begin{aligned} Q_{3} &=\text{Value of }\bigg(\dfrac{3(n+1)}{4}\bigg)^{th} \text{ observation}\\ &=\text{Value of }\bigg(\dfrac{3(15+1)}{4}\bigg)^{th} \text{ observation}\\ &= \text{Value of }\big(12\big)^{th} \text{ observation}\\ &=13 \text{ days}. \end{aligned} $$
Thus, lower $75$ % of the patients had length of stay in the hospital less than or equal to $13$ days.

Quartile Deviation

The quartile deviation is

$$ \begin{aligned} QD &= \frac{Q_3 - Q_1}{2}\\ &= \frac{13 - 9}{2}\\ &= \frac{4}{2}\\ &=2 \text{ days}. \end{aligned} $$
Coefficient of Quartile Deviation

The coefficient of quartile deviation is

$$ \begin{aligned} \text{Coefficient of }QD &= \frac{Q_3 - Q_1}{Q_3+Q_1}\\ &= \frac{13 - 9}{13+9}\\ &= \frac{4}{22}\\ &=0.1818. \end{aligned} $$

Quartile deviation for ungrouped data Example 2

Blood sugar level (in mg/dl) of a sample of 20 patients admitted to the hospitals are as follows:

75,89,72,78,87, 85, 73, 75, 97, 87, 84, 76,73,79,99,86,83,76,78,73.

Find the Quartiles deviation and coefficient of QD.

Solution

The formula for $i^{th}$ quartile is

$Q_i =$ Value of $\bigg(\dfrac{i(n+1)}{4}\bigg)^{th}$ observation, $i=1,2,3$

where $n$ is the total number of observations.

Arrange the data in ascending order

72, 73, 73, 73, 75, 75, 76, 76, 78, 78, 79, 80, 82, 83, 84, 85, 86, 87, 97, 99

First Quartile $Q_1$

The first quartle $Q_1$ can be computed as follows:

$$ \begin{aligned} Q_{1} &=\text{Value of }\bigg(\dfrac{1(n+1)}{4}\bigg)^{th} \text{ observation}\\ &=\text{Value of }\bigg(\dfrac{1(20+1)}{4}\bigg)^{th} \text{ observation}\\ &= \text{ Value of }\big(5.25\big)^{th} \text{ observation}\\ &= \text{Value of }\big(5\big)^{th} \text{ obs.}\\ &\quad +0.25 \big(\text{Value of } \big(6\big)^{th}\text{ obs.}-\text{Value of }\big(5\big)^{th} \text{ obs.}\big)\\ &=75+0.25\big(75 -75\big)\\ &=75 \text{ mg/dl}. \end{aligned} $$

Thus, lower $25$ % of the patients had blood sugar level less than or equal to $75$ mg/dl.

Third Quartile $Q_3$

The third quartile $Q_3$ can be computed as follows:

$$ \begin{aligned} Q_{3} &=\text{Value of }\bigg(\dfrac{3(n+1)}{4}\bigg)^{th} \text{ observation}\\ &=\text{Value of }\bigg(\dfrac{3(20+1)}{4}\bigg)^{th} \text{ observation}\\ &= \text{ Value of }\big(15.75\big)^{th} \text{ observation}\\ &= \text{Value of }\big(15\big)^{th} \text{ obs.}\\ &\quad +0.75 \big(\text{Value of } \big(16\big)^{th}\text{ obs.}-\text{Value of }\big(15\big)^{th} \text{ obs.}\big)\\ &=84+0.75\big(85 -84\big)\\ &=84.75 \text{ mg/dl}. \end{aligned} $$

Thus, lower $75$ % of the patients had blood sugar level less than or equal to $84.75$ mg/dl.

Quartile Deviation

The quartile deviation is

$$ \begin{aligned} QD &= \frac{Q_3 - Q_1}{2}\\ &= \frac{84.75 - 75}{2}\\ &= \frac{9.75}{2}\\ &=4.875\text{ mg/dl}. \end{aligned} $$
Coefficient of Quartile Deviation

The coefficient of quartile deviation is

$$ \begin{aligned} \text{Coefficient of }QD &= \frac{Q_3 - Q_1}{Q_3+Q_1}\\ &= \frac{84.75 - 75}{84.75+75}\\ &= \frac{9.75}{159.75}\\ &=0.061. \end{aligned} $$

Quartile Deviation for ungrouped data Example 3

The following data are the heights, correct to the nearest centimeters, for a group of children:

126, 129, 129, 132, 132, 133, 133, 135, 136, 137, 
137, 138, 141, 143, 144, 146, 147, 152, 154, 161 

Find the quartile deviation and coefficient of QD for the given data.

Solution

The formula for $i^{th}$ quartile is

$Q_i =$ Value of $\bigg(\dfrac{i(n+1)}{4}\bigg)^{th}$ observation, $i=1,2,3$

where $n$ is the total number of observations.

Arrange the data in ascending order

126, 129, 129, 132, 132, 133, 133, 135, 136, 137, 137, 138, 141, 143, 144, 146, 147, 152, 154, 161

First Quartile $Q_1$

The first quartle $Q_1$ can be computed as follows:

$$ \begin{aligned} Q_{1} &=\text{Value of }\bigg(\dfrac{1(n+1)}{4}\bigg)^{th} \text{ observation}\\ &=\text{Value of }\bigg(\dfrac{1(20+1)}{4}\bigg)^{th} \text{ observation}\\ &= \text{ Value of }\big(5.25\big)^{th} \text{ observation}\\ &= \text{Value of }\big(5\big)^{th} \text{ obs.}\\ &\quad +0.25 \big(\text{Value of } \big(6\big)^{th}\text{ obs.}-\text{Value of }\big(5\big)^{th} \text{ obs.}\big)\\ &=132+0.25\big(133 -132\big)\\ &=132.25 \text{ cm}. \end{aligned} $$

Thus, lower $25$ % of the patients had height less than or equal to $132.25$ cm.

Third Quartile $Q_3$

The third quartile $Q_3$ can be computed as follows:

$$ \begin{aligned} Q_{3} &=\text{Value of }\bigg(\dfrac{3(n+1)}{4}\bigg)^{th} \text{ observation}\\ &=\text{Value of }\bigg(\dfrac{3(20+1)}{4}\bigg)^{th} \text{ observation}\\ &= \text{ Value of }\big(15.75\big)^{th} \text{ observation}\\ &= \text{Value of }\big(15\big)^{th} \text{ obs.}\\ &\quad +0.75 \big(\text{Value of } \big(16\big)^{th}\text{ obs.}-\text{Value of }\big(15\big)^{th} \text{ obs.}\big)\\ &=144+0.75\big(146 -144\big)\\ &=145.5 \text{ cm}. \end{aligned} $$

Thus, lower $75$ % of the patients had height less than or equal to $145.5$ cm.

Quartile Deviation

The quartile deviation is

$$ \begin{aligned} QD &= \frac{Q_3 - Q_1}{2}\\ &= \frac{145.5 - 132.25}{2}\\ &= \frac{13.25}{2}\\ &=6.625\text{ cm}. \end{aligned} $$
Coefficient of Quartile Deviation

The coefficient of quartile deviation is

$$ \begin{aligned} \text{Coefficient of }QD &= \frac{Q_3 - Q_1}{Q_3+Q_1}\\ &= \frac{145.5 - 132.25}{145.5+132.25}\\ &= \frac{13.25}{277.75}\\ &=0.0477. \end{aligned} $$

Quartile Deviation for ungrouped data Example 4

The following measurement were recorded for the drying time in hours, of a certain brand of latex paint.

3.4 2.5 4.8 2.9 3.6 2.8 3.3 5.6 
3.7 2.8 4.4 4.0 5.2 3.0 4.8.

Compute Quartile deviation and coefficient of QD for the above data.

Solution

The formula for $i^{th}$ quartile is

$Q_i =$ Value of $\bigg(\dfrac{i(n+1)}{4}\bigg)^{th}$ observation, $i=1,2,3$

where $n$ is the total number of observations.

Arrange the data in ascending order

2.5, 2.8, 2.8, 2.9, 3, 3.3, 3.4, 3.6, 3.7, 4, 4.4, 4.8, 4.8, 5.2, 5.6

First Quartile $Q_1$

The first quartle $Q_1$ can be computed as follows:

$$ \begin{aligned} Q_{1} &=\text{Value of }\bigg(\dfrac{1(n+1)}{4}\bigg)^{th} \text{ observation}\\ &=\text{Value of }\bigg(\dfrac{1(15+1)}{4}\bigg)^{th} \text{ observation}\\ &= \text{Value of }\big(4\big)^{th} \text{ observation}\\ &=2.9 \text{ hours}. \end{aligned} $$

Thus, lower $25$ % of the drying time is less than or equal to $2.9$ hours.

Third Quartile $Q_3$

The third quartile $Q_3$ can be computed as follows:

$$ \begin{aligned} Q_{3} &=\text{Value of }\bigg(\dfrac{3(n+1)}{4}\bigg)^{th} \text{ observation}\\ &=\text{Value of }\bigg(\dfrac{3(15+1)}{4}\bigg)^{th} \text{ observation}\\ &= \text{Value of }\big(12\big)^{th} \text{ observation}\\ &=4.8 \text{ hours}. \end{aligned} $$
Thus, lower $75$ % of the drying time is less than or equal to $4.8$ hours.

Quartile Deviation

The quartile deviation is

$$ \begin{aligned} QD &= \frac{Q_3 - Q_1}{2}\\ &= \frac{4.8 - 2.9}{2}\\ &= \frac{1.9}{2}\\ &=0.95 \text{ hours}. \end{aligned} $$
Coefficient of Quartile Deviation

The coefficient of quartile deviation is

$$ \begin{aligned} \text{Coefficient of }QD &= \frac{Q_3 - Q_1}{Q_3+Q_1}\\ &= \frac{4.8 - 2.9}{4.8+2.9}\\ &= \frac{1.9}{7.7}\\ &=0.2468. \end{aligned} $$

Quartile Deviation for ungrouped data Example 5

The rice production (in Kg) of 10 acres is given as: 1120, 1240, 1320, 1040, 1080, 1720, 1600, 1470, 1750, and 1885. Find the quartile deviation and coefficient of QD for the given data.

Solution

The formula for $i^{th}$ quartile is

$Q_i =$ Value of $\bigg(\dfrac{i(n+1)}{4}\bigg)^{th}$ observation, $i=1,2,3$

where $n$ is the total number of observations.

Arrange the data in ascending order

1040, 1080, 1120, 1240, 1320, 1470, 1600, 1720, 1750, 1885

First Quartile $Q_1$

The first quartle $Q_1$ can be computed as follows:

$$ \begin{aligned} Q_{1} &=\text{Value of }\bigg(\dfrac{1(n+1)}{4}\bigg)^{th} \text{ observation}\\ &=\text{Value of }\bigg(\dfrac{1(10+1)}{4}\bigg)^{th} \text{ observation}\\ &= \text{ Value of }\big(2.75\big)^{th} \text{ observation}\\ &= \text{Value of }\big(2\big)^{th} \text{ obs.}\\ &\quad +0.75 \big(\text{Value of } \big(3\big)^{th}\text{ obs.}-\text{Value of }\big(2\big)^{th} \text{ obs.}\big)\\ &=1080+0.75\big(1120 -1080\big)\\ &=1110 \text{ Kg}. \end{aligned} $$

Thus, lower $25$ % of the patients had rice production less than or equal to $1110$ Kg.

Third Quartile $Q_3$

The third quartile $Q_3$ can be computed as follows:

$$ \begin{aligned} Q_{3} &=\text{Value of }\bigg(\dfrac{3(n+1)}{4}\bigg)^{th} \text{ observation}\\ &=\text{Value of }\bigg(\dfrac{3(10+1)}{4}\bigg)^{th} \text{ observation}\\ &= \text{ Value of }\big(8.25\big)^{th} \text{ observation}\\ &= \text{Value of }\big(8\big)^{th} \text{ obs.}\\ &\quad +0.25 \big(\text{Value of } \big(9\big)^{th}\text{ obs.}-\text{Value of }\big(8\big)^{th} \text{ obs.}\big)\\ &=1720+0.25\big(1750 -1720\big)\\ &=1727.5 \text{ Kg}. \end{aligned} $$

Thus, lower $75$ % of the patients had rice production less than or equal to $1727.5$ Kg.

Quartile Deviation

The quartile deviation is

$$ \begin{aligned} QD &= \frac{Q_3 - Q_1}{2}\\ &= \frac{1727.5 - 1110}{2}\\ &= \frac{617.5}{2}\\ &=308.75\text{ Kg}. \end{aligned} $$
Coefficient of Quartile Deviation

The coefficient of quartile deviation is

$$ \begin{aligned} \text{Coefficient of }QD &= \frac{Q_3 - Q_1}{Q_3+Q_1}\\ &= \frac{1727.5 - 1110}{1727.5+1110}\\ &= \frac{617.5}{2837.5}\\ &=0.2176. \end{aligned} $$

Conclusion

In this tutorial, you learned about formula for quartile deviation for ungrouped data and how to calculate quartile deviation and coefficient of quartile deviation for ungrouped data. You also learned about how to solve numerical problems based on quartile deviation for ungrouped data.

To learn more about other descriptive statistics measures, please refer to the following tutorials:

Descriptive Statistics

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